Roots of Unity

Roots of unity:

For $n\in\mathbb{N}$ , the $n$ -th roots of unity are the complex solutions of $z^n=1$ .

They are given by $z_k = e^{i\frac{2\pi k}{n}}$ for $k=0,\dots,n-1$ .

Equivalently, these are the $n$ equally spaced points on the unit circle.

Polygonal approximation of an arc (lecture note motivation):

Let $x\in[0,2\pi]$ and $n\in\mathbb{N}$ . Define points on the unit circle by $A_k := e^{i\frac{kx}{n}}$ for $k=0,\dots,n$ . Let $\ell_n$ be the length of the polygonal chain through $A_0,A_1,\dots,A_n$ : $\ell_n := \sum_{k=1}^n |A_k-A_{k-1}|$ .

Then $\ell_n = 2n\sin\!\big(\frac{x}{2n}\big)$ and $\lim_{n\to\infty}\ell_n = x$ .

How to use the applet

Move the slider $n$ to choose how many roots of unity are displayed.
The blue points are the $n$ -th roots of unity

z_k = e^{i\frac{2\pi k}{n}}, \qquad k=0,\dots,n-1.

The blue polygon connects the points in order, showing that the roots form a regular $n$ -gon inscribed in the unit circle.
The roots are equally spaced: the angle between consecutive roots is $\frac{2\pi}{n}$ .
Increasing $n$ makes the polygon look closer and closer to the circle (this is the geometric idea behind approximating circular quantities by polygons).

Loading graph…

Hold Shift + scroll to zoom
Hold Shift + drag to move
Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Figure 5.5 page 89

This applet visualizes the $n$ -th roots of unity, i.e. the solutions of $z^n=1$ . For a chosen integer $n$ (slider nSlider), the points $z_k = e^{i\frac{2\pi k}{n}}$ for $k=0,\dots,n-1$ are displayed on the unit circle as equally spaced points, and the blue polygon connects them to emphasize that the roots of unity form a regular $n$ -gon inscribed in the unit circle.

In the code, the roots are created as points on the unit circle with coordinates $\big(\cos(\frac{2\pi k}{n}),\sin(\frac{2\pi k}{n})\big)$ , using the expressions Math.cos((TAU * k) / n) and Math.sin((TAU * k) / n) (with TAU = 2 * Math.PI). Concretely, the applet creates an array of points roots.

        const roots: JXG.Point[] = [];
        for (let k = 0; k < MAX_N; k++) {
          roots.push(
            createPoint(
              board,
              [
                () => Math.cos((TAU * k) / Math.floor(nSlider.Value())),
                () => Math.sin((TAU * k) / Math.floor(nSlider.Value())),
              ],
              {
                ...
              },
              ...
            )
          );
        }